Let \((G, l)\)  be a metric graph with the length defined by a positive vector \(l\in \mathbb {R}^{E}_{>0}\) . A notion of the \(C\) -isoperimetric condition on it was introduced in [1]} as a discrete version of the isoperimetric inequality on Riemannian surface. Set the boundary for any \(U\subset V\)  as
\(\partial U=\lbrace ij\in E:i\in U,\text{ and }j\notin U\rbrace ,\) 
